1 . 我国古代数学名著《九章算术》,将底面为矩形且有一条侧棱垂直于底面的四棱锥称为“阳马”.如图所示,在长方体
中,已知
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/31/064926f8-580d-47cb-ba38-0fa73946e3aa.png?resizew=134)
(1)求证:四棱锥
是一个“阳马”,并求该“阳马”的体积;
(2)求该“阳马”
的外接球的表面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f121eabff3c62c1a196d9ca5f6f83f0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8d927585a17c2e98ef7d5a9589a26ac.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/31/064926f8-580d-47cb-ba38-0fa73946e3aa.png?resizew=134)
(1)求证:四棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fec35c2182c5e0c80b766adceb058e5f.png)
(2)求该“阳马”
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fec35c2182c5e0c80b766adceb058e5f.png)
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2 . 某广场内设置了一些石凳供大家休息,这些石凳是由正方体截去八个一样的四面体得到的,如图所示,若被截正方体的棱长是60cm.
(2)为了美观工人准备将石凳的表面进行粉刷,已知每平方米造价50元,请问粉刷一个石凳需要多少钱?(精确到0.1元)
(2)为了美观工人准备将石凳的表面进行粉刷,已知每平方米造价50元,请问粉刷一个石凳需要多少钱?(精确到0.1元)
您最近一年使用:0次
2023-10-22更新
|
583次组卷
|
8卷引用:上海市青浦区朱家角中学2023-2024学年高二上学期期末考试数学试题
上海市青浦区朱家角中学2023-2024学年高二上学期期末考试数学试题黑龙江省实验中学2022-2023学年高一下学期期中数学试题内蒙古自治区乌兰察布市衡水卓远中学2022-2023学年高一下学期6月月考数学试题辽宁省沈阳市翔宇中学2022-2023学年高一下学期第二次月考测试数学试题上海市七宝中学2023-2024学年高二上学期10月月考数学试题上海市七宝中学2023-2024学年高二上学期9月月考数学试题(已下线)模块三 专题5 大题分类练(空间几何体表面积和体积)(人教A版)(已下线)专题03 简单几何体的表面积和体积-《知识解读·题型专练》(人教A版2019必修第二册)
解题方法
3 . 如图,在多面体
中,
是四边形
的外接圆的直径,
是
与
的交点,
,
.四边形
是直角梯形,
,
平面
,
.
(1)求证:
平面
;
(2)求多面体
的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9165d9bfbb0f0d19eb482c2a4c1b29b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e735a28578ba191da6d4f3b0f8e8729.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5640c531110986a503de677715770e62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ecc1cb55a57dde481f8dd07ab150676.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8139d9fd5c670c91aa7dc485366dd1e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8d2d217e9bcd059908f117dfc4d4259.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4592d3d223b4f8c93c9308afadfa79b3.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/8/3/ba2c66a8-559a-42e7-950a-89bb8b177514.png?resizew=133)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8197bf06d017950c85c3ba6a291c095e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87c0bfeadcf17b2a45896071f07a4a5a.png)
(2)求多面体
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9165d9bfbb0f0d19eb482c2a4c1b29b7.png)
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解题方法
4 . 如图I为某同学搭建的立体几何模型,相关性质如图描述,其侧面展开图如图II所示.图I中,圆锥
的半径为3,体积为12π. 在等腰
(可近似看作与扇形KUN重合)中,
.中间圆柱展开图可看作正方形.圆柱J-G中,半径为3,体积为45π.侧面非阴影部分的圆边共占20%.设圆O所在平面为
,圆G所在平面为
,各立方体平稳放置,回答以下问题:
.
(2)试求K到G的距离及阴影部分面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/181928a80db33cfb7b903f50ebad01f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c27f6dcfcbc956cab53d53f39d5c47d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c11a3684becedd099419cc3fc019373.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce52a7419783b3542d19b755e2cb028d.png)
(2)试求K到G的距离及阴影部分面积.
您最近一年使用:0次
解题方法
5 . 如图,在正四棱柱
中,
,
∥平面MAC.
(1)证明:M是
的中点;
(2)若正四棱柱的外接球的体积是
,求该正四棱柱的表面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92535536bd3c2761724fd058427f95a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fe734023d4e70010a6b2cc3267cb86e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/8/5/e7673eec-8fe4-47f5-a514-8bc8c82a4303.png?resizew=125)
(1)证明:M是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22adbc0da438220f9cace11b629d799b.png)
(2)若正四棱柱的外接球的体积是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eefdcd39676298f214654051fc51ba92.png)
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2023-07-28更新
|
572次组卷
|
2卷引用:安徽省宣城市2022-2023学年高一下学期期末调研测试数学试卷
解题方法
6 . 如图所示的多面体
中,
是等边三角形,平面
平面
,平面
平面
.
(1)求证:
//平面
;
(2)若
,求多面体
的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9142a8490de14a87eda628ffa7e28982.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/309426bb62d195dc56d2ec3fa3f4e2a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17580410bf63dba4fe164265afaac4cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39282bdf319f30d7bc261e2e3ab3b1e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/8/1/e2a4e0dd-7873-4744-a000-1dea6bc7f5c2.png?resizew=181)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9142a8490de14a87eda628ffa7e28982.png)
您最近一年使用:0次
7 . 在如图所示的几何体中,底面
是正方形,四边形
是直角梯形,
,
,平面
平面
,
分别为
的中点,
,
.
平面
;
(2)求多面体
的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c9854839f8f7fe792cd83cf3aa8b093.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b137f02d1323fe46ce853f662542d2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/160ccfe256fc5347daa5e4cc26719512.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e675e59fa66ecdf14ba695e5e649222.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fadb75d3985c7cf99cc8e794ae2d7b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86a8a140610df89623519116d9e9697c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca5a9a04de2ddcec2b2799ab5476f2c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa3d5158341e73d86b6308be42a22fd1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1fd975b889bfe7ddcec0de56b6f23ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c9854839f8f7fe792cd83cf3aa8b093.png)
(2)求多面体
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bab155dd2cd44b7301963056f9b0444b.png)
您最近一年使用:0次
2023-07-27更新
|
266次组卷
|
2卷引用:福建省三明市2022-2023学年高一下学期期末质量检测数学试题
8 . 如图,在三棱柱
中,
,
,
平面
.
平面
;
(2)若点
在棱
上,当
的面积最小时,求三棱锥
外接球的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f90e17995e2f71e297d94ae51c7e5b1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43732729894297552d9210f41a634769.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e56fdf217165748fafe938b64fa08179.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9edc50f7febbc2d5d8dcdc23a3630a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e16f65c3a318220c2f5baac171bbb61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a211ad5a06b505b8365a62c1946f3cb7.png)
(2)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11ddc92d84d188c66b435664a7e7b5a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcb49df05f2e31d005735c3f14a21d30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6eedadb500cb9faa9de59d0cfdf338c0.png)
您最近一年使用:0次
名校
解题方法
9 . 如图,在多面体
中,平面
平面
,四边形
为菱形,
,底面
为直角梯形,
,
,
.
(1)证明:
;
(2)若
,求多面体
的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9165d9bfbb0f0d19eb482c2a4c1b29b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cf9a6db3571fa57bfa2d5e4d44c51b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b32c05247f6998d7a70d31d13be4148c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b32c05247f6998d7a70d31d13be4148c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f8c595bc64da14b0f78019c54da608d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10df84d553a8826a7ce9bff4bf0d95b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/080db3af81b29ed10144a1c2e2a4fb8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70e8b3dc5a29951f630424fd7865b6c4.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/27/902d6c09-c149-421f-8f49-71b6124eeb4d.png?resizew=185)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c252a0fe067d434a2b5aeac011b9914.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ced06b71073e1bb777f326f06016ce17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9165d9bfbb0f0d19eb482c2a4c1b29b7.png)
您最近一年使用:0次
名校
10 . 等腰梯形
中,
,
,
.若点
、
均在
上,且
.如图(一)所示,沿
将
折起,沿
将
折起,使
、
两点重合为
.
(1)若
,如图(二)所示,求证:平面
平面
;
(2)若
,
为
中点,当
与
重合于
时,如图(三)所示,求
与平面
所成角的余弦值;
(3)请设计一个翻折方案使四棱锥
的外接球半径为
,证明你的结论,并求此方案下的
的长度及
的大小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a11029ca6b4b9e7f777af0280cf163c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b040eb31b0b7073ad3ffa8bd7968d187.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/037b342a682cbd4241855a243da3c016.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04f8eebda19eded2b059774a8c2666c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6830ebecddbd9759be626289c408e4f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/27/ba897f14-f9d7-44dc-b819-8c1cfd0adc02.png?resizew=459)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27aa17bad024a9361bd0a679e10f70ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6bec37dca00db5f4512ce70f16ceb20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3747e528a1e8d45668ccf835c0175a73.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f4aca5534bce25acaeb7379deed8f8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18e5ef91fb27dd684a27ae7f1993cfba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ef796b46e68fe77b117ff0483d2370c.png)
(3)请设计一个翻折方案使四棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ff027309f3108559e6b3915158a3867.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b3b0b11a80e8b107e55534d7fda9f2b.png)
您最近一年使用:0次